Problem 25 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\frac{2 x}{x-2}=5+\frac{4 x^{2}}{x-2}$$

Step-by-Step Solution

Answer

The solutions are \(\frac{5}{4}\) and \(-2.\)

Step 1: Eliminate the Denominator

Multiply both sides of the equation by \(x-2\) to eliminate the denominator. This results in \[2x = 5(x-2) + 4x^2.\]

Step 2: Distribute and Simplify

Distribute 5 on the right-hand side of the equation and simplify: \[2x = 5x - 10 + 4x^2.\]

Step 3: Move All Terms to One Side

Rearrange the equation so that all terms are on one side: \[4x^2 + 3x - 10 = 0.\]

Step 4: Factor the Quadratic Equation

Factor the quadratic equation \[4x^2 + 3x - 10 = 0.\] The factored form is \[(4x - 5)(x + 2) = 0.\]

Step 5: Solve for x

Set each factor equal to zero and solve for \(x\): \(4x - 5 = 0\) leads to \((4x = 5, x = \frac{5}{4}).\) \(x + 2 = 0\) leads to \(x = -2.\)

Key Concepts

Rational Equations

Rational equations are equations that involve fractions whose numerators and/or denominators contain polynomials. Understanding how to solve these equations is crucial because they often appear in algebra and beyond. To solve a rational equation, follow these steps:

• Find a common denominator
• Eliminate the denominators by multiplying through by that common denominator
• Simplify and solve the resulting polynomial equation

In our problem, we have: \(\frac{2 x}{x-2}=5+\frac{4 x^{2}}{x-2}\). By eliminating the denominator, we simplify the equation and make it easier to solve.

Quadratic Equations

A quadratic equation is any equation of the form \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants. These equations are fundamental in algebra and appear frequently in mathematics. To solve a quadratic equation, you can:

• Factorize it, if possible
• Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Complete the square

For our example, after removing the fractions and simplifying, we ended up with the quadratic equation \(4x^2 + 3x - 10 = 0\). This form is much easier to handle with standard methods.

Factoring

Factoring is a method to write an expression as a product of its factors. Factoring quadratic equations is particularly useful because it allows us to find the roots of the equation easily. Here’s a quick guide:

• Look for common factors
• Use special product rules (difference of squares, perfect square trinomials)
• Split the middle term, if necessary

In our step-by-step solution, we factored the quadratic equation \(4x^2 + 3x - 10 = 0\) to \((4x - 5)(x + 2) = 0\). By setting each factor to zero, we found the solutions: \(x = \frac{5}{4}\) and \(x = -2\). Factoring made finding the solution much more manageable.